Analytic Verification of the Droplet Picture in the Two-Dimensional Ising Model

It is widely accepted that the free energy F(H) of the two-dimensional Ising model in the ferromagnetic phase T<T _c has an essential branch cut singularity at the origin H=0. The phenomenological droplet theory predicts that near the cut drawn along the negative real axis H<0, the imaginary part of the free energy per lattice site has the form ImFexp(±i)|H|]=±B|H|exp(–A/|H|) for small |H|. We verify this prediction in analytical perturbative transfer matrix calculations for the square lattice Ising model for all temperatures 0<T<T _c and arbitrary anisotropy ratio J ₁/J ₂. We obtain an expression for the constant A which coincides exactly with the prediction of the droplet theory. For the amplitude B we obtain B=M/18, where M is the equilibrium spontaneous magnetization. In addition we find discrete-lattice corrections to the above mentioned phenomenological formula for ImF, which oscillate in H ^–1.